Thermodynamic properties of inhomogeneous fluids

Physica 12lA (1983) 399429 North-Holland

THERMODYNAMIC

Publishing Co.

PROPERTIES

OF INHOMOGENEOUS

FLUIDS

Michel BOITEUX Laboratoire

d’Ultrasons,

VniversitP Pierre et Marie

Curie,

75230 Paris Cedex

05, France

and John KERINS Department

of Chemical Engineering

and Materials Science, MN 55455, USA

University

of Minnesota,

Minneapolis,

Received 15 April 1983

A general method, the method of variation under extension, is presented for expressing the thermodynamic properties of an inhomogeneous fluid as functionals of the local number density, when given a density functional for the total thermodynamic grand potential of the fluid. The method is demonstrated in detail for the van der Waals square-gradient density functional and for the nonlocal density functional which arises in the theory of fluids with long-ranged pair potentials or in the mean-field theory of penetrable-sphere models. As specific examples, we consider the planar and spherical interface between two fluid phases, the line of contact of three fluid phases, the contact line between two surface phases and the planar interface between a solid and fluid.

1. Introduction

A simple example of an inhomogeneous equilibrium system is a sample of a single-component, subcritical fluid, in which a planar interface separates coexisting liquid and vapor phases, called phases b and LX,respectively. The thermodynamic properties of this inhomogeneous system depend on the properties of the coexisting bulk phases and on those of the planar interface’). For example, we must consider both the bulk-phase pressure p and the interfacial tension aOLfl when analyzing the mechanical work which can be done on the sample. On a microscopic level, this inhomogeneous fluid is most readily characterized by the density profile. With the coordinate x measuring distance perpendicular to the c$ interface (fig. la), the fluid density p(x) differs appreciably from its value in the c1or fl bulk phase only in the interface. A reasonable microscopic theory of the interface should provide a clear prescription for calculating interfacial thermodynamic properties, such as the tension @, given the appropriate microscopic information, say as contained in p(x). In fact in any microscopic theory of inhomogeneous fluid, one goal is to determine the thermodynamic properties 037%4371/83/000&0000/$03.00

0

1983 North-Holland.

400

M. BOITEUX

AND

J. KERINS

b

a

Fig. I. (n) A planar r/i interface: (b) a spherical r/j Interface; (c) a system of three bulk phases. 1. /j and T. which arc separated by two planar interfaces; (d) a system in which three bulk phases. 2. /j and ;‘, and three planar interfaces. r/l. z: and [P;. meet along a straight line of contact.

of the inhomogeneities sample of equilibrium

given their microscopic fluid as inhomogeneous

structure. We define a macroscopic if we can find two separated regions

in the sample, which contain isotropic fluid phases with a localized region of inhomogeneous fluid between the phases. For example, in contrast to the planar Z/I interface. we could imagine a bubble of s( phase separated from the surrounding /I phases by a spherical C$ interface (fig. lb); in this case the density p depends only on the radial distance r as measured from the center of the bubble. Or we could consider a fluid sample in which we find three regions, each filled with one of three distinct but coexisting bulk phases called 2, /I and 7. Now we might find the three phases separated by planar X/I and /?I) interfaces, with the fluid densities depending only on the coordinate s (fig. Ic). or, alternatively, we might find the three phases separated by three planar interfaces (shown as the planes xfi, /5; and r;~ in fig. Id), which meet along a straight line of contact’). In the latter case, where the fluid densities depend on coordinates .I-’ and .r2 perpendicular to the contact line, the line tension t is a thermodynamic quantity which should be determined in a microscopic theory, just as should the interfacial tensions a”‘. G’~‘,and ox,. In this paper we present a method for isolating and evaluating the thermodynamic properties of the inhomogeneities in an inhomogeneous fluid in the case that the equilibrium grand potential of the fluid is given by a density functional’). which is minimal with respect to variations in the local fluid densities. That is. we choose the density functional formalism as the framework for our microscopic theory of inhomogeneous fluid. The key to the method is an examination of the change in the grand potential Q of the fluid sample when the size of the sample

THERMODYNAMIC

is increased, practice

PROPERTIES

i.e., an examination determined

Accordingly,

while we do present

method

of the variation

the exact density functional

52 is usually

of variation

by example.

under

briefly

401

FLUIDS

extension.

for an inhomogeneous

by minimizing

under extension,

In an earlier

OF INHOMOGENEOUS

Of course

fluid is not known,

some approximate

density

some of the general

paper4) we were able to isolate

and

functional.

formulas

our main effort is to demonstrate

in

for the

the method

the line tension

of a

straight contact line by applying Noether’s theorem in the case of a van der Waals density functional. With the present method of variation under extension, we can not only reproduce this earlier result, but can also produce similar results for more general density functionals to which Noether’s theorem is not applicable. In section 2 we examine the variation under extension of the van der Waals density functional for a planar interface. In this simple case, we compute the variation in the grand potential, 652, by resealing directly the length coordinate. We introduce in section 3 a more elegant approach to determining 652 for a van der Waals functional through a functional differentiation with respect to the metric tensor, and then we apply the method in several different cases. This technique of functional differentiation is extended in section 4 to nonlocal density functionals, we discuss

and again the method is illustrated by several examples. In section 5 some of the limitations and further implications of the method of

variation under extension. results to classical results

2. The conventional

We also comment on the relationship in the theory of continuum mechanics.

of our present

analysis of variation under extension

Let us consider a single-component fluid at a fixed subcritical temperature T and at the associated coexistence chemical potential p(T), in which vapor or c1Phase is separated

from liquid or /I phase by a planar

interface.

We restrict our attention

to a macroscopic sample of the fluid contained in an imaginary rectangular cylinder, which has two faces of area A parallel to the interface and four edges of length 2L perpendicular to the interface (fig. 2). If the total grand potential of this sample is fi(A, L), then in a classical Gibbsian thermodynamic analysis, the grand potential per unit area or reduced grand potential Q(L) is L?(L)=7

S2”M L)

= -- 2pL + #‘p + 0(1/L),

with p the common value of the pressure in bulk phases a and ,!), and oUB the interfacial tension. Terms of order (l/L) are expected because of the fine size of the sample; even with L large, the fluid near the top or bottom of our imaginary sample container may be slightly different from the fluid in bulk a or p phase, respectively, because of the interface inside the container. Now imagine that we

M. BOITEUX

402

AND

-L-CL

Fig. 2. A sample area A.

extend

of the two-phase

our sample

J. KERINS

1

system in a rectangular

volume

by increasing

cylinder

the sample

from which we can conclude f+=lim

i

L-,X,

a(L)--lim?

C-10 t

2pL)c + 0(1/L)

2L and cross-sectional

height by an amount

top and bottom (0 < c 4 1) at fixed cross-sectional under this extension is 652, 652 = s2(L + EL) - Q(L) = (-

of height

area A. The variation

tL on

of Sz

)

(2)

that

1

(3)

Thus r~‘flis given by the asymptotic difference of the grand potential per unit area, Q(L), and the variation under extension divided by the scale of the variation, 60/t. If this two-phase system is described on a microscopic level by the well-known van der Waals or gradient theory of interface@‘), then the reduced grand potential

is taken

to be a functional

W)=Q[p;Ll=

j

of the equilibrium

density

dx{$(p)fc(p)(~~j.

profile p(x);

(4)

with c a positive function of p. The profile p(x) affords a minimum to the functional and necessarily satisfies the associated Euler-Lagrange equation.

(5) subject to the boundary the bulk GLor /I phases,

conditions that p(x) approach p’ or pB, the density in as x goes to + co, respectively. In the van der Waals

THERMODYNAMIC

PROPERTIES

OF INHOMOGENEOUS

FLUIDS

functional (4) the local density of grand potential is the sum of two the first tj(p(x)), which is the grand potential density in a uniform constrained to have density p(x) at the given T and ,u, and the c(p)(dp/dx)*, which is a positive contribution due to the inhomogeneity. goes to co, dp/dx goes to zero and $(p) goes to - p, the negative equilibrium pressure. For the given van der Waals functional (4), we could directly utilize the thermodynamic behavior (1) of Q(L) to write

403

pieces, system second As 1x1 of the known

@ = lim [Q(L) + 2pL] L-m

.

(6)

-02

In this integral expression for the tension aafi the integrand, although it does explicitly require the bulk-phase pressure p, is non-zero only in the interface. But now instead of this direct method, let us apply to the functional (4) the prescription for determining rr@embodied in the asymptotic difference formula (3) which follows from the method of variation under extension. We first extend the system by 2.5L in height, and then make a transformation of coordinates, (1 + c)y = x, to recover an integral with upper and lower limits of + L, respectively. L(l+G

s2(L + EL) =

dx

j-

-L(l

with b(x) = p(x + a). #iqx)=p(x)+

(

$

{W+cCjo)‘)

+c)

68 is now calculated by expanding p’(x) about p(x),

>

cx+O(c2),

(8)

with the result that

(9)

404

M. BOITEUX

Given

the asymptotic

The integral

difference

expression

the equilibrium

profile

AND J. KERINS

formula

(3) for a’” we then have

for a”’ can be further in the form

in the integrand of (IO). and formula for the tension,

then

simplified

integrating

by parts

by first using eq. (5) for

to find a well-known

(12)

While we have again expressed only in the interface,

the present

cr’fi as the integral integral

of a function

which is non-zero

(12) differs in several important

respects

from the earlier integral formula (6). The integrand in (12) is simpler than that of (6) and could be more convenient for numerical evaluation of crXB.Indeed, the density latter integral expression (12) for 0 ‘1 is valid only if p(s) is the equilibrium profile; the difference in aXB as calculated from (6) and (12) could be used as an estimate of the numerical error. More importantly, it is obvious from (12) that rrrB > 0, in accord with thermodynamic stability, but no such conclusion is readily apparent

from (6) since the structure

of G(p) is not specified.

The method of variation under extension may be applied to quantities other than the grand potential per unit area. For example, if N is the total number of particles

in our sample,

then choosing

the Gibbs

dividing

surface at x = 0 we may

write

where

r is the adsorption

N =

at the interface.

On a microscopic

J

d.v p(s)

and by following

the steps of (7))( 12) we deduce

level we have

(14)

that

THERMODYNAMIC

PROPERTIES

OF INHOMOGENEOUS

FLUIDS

405

From a thermodynamic point of view (7@ is the quantity of fundamental interest. Because we started with the grand potential of the system, o@(T, p) is the thermodynamic potential for the interfacial phase, i.e., dazP=s”fidT+r

(16)

dP,

where s’fi is the interfacial entropy density. Furthermore, a@ is independent of the choice of Gibbs dividing surface whereas r and s@ are not. In terms of the microscopic description, if p,(x) is a solution of the profile eq. (5), then so is p*(x) = p,(x + x,,), x,, a constant; but whereas the surface tension (12) of the two solutions will be the same, the adsorptions will be different r, # Tz for the same choice x = 0 of the dividing surface. Moreover, only in the integral expression (12) for alp did we explicitly use to advantage the fact that p(x) is an equilibrium density profile. On a macroscopic or thermodynamic level, the fundamental equation for the method of variation under extension as applied to the planar interface is (3). The method is useful, however, only because for the given van der Waals functional (4) the variation 6Q can be explicitly calculated. The key step in this calculation of 652 was resealing the length coordinate (7) which effectively transforms a variation at the boundaries of the system into a variation over the interior of the system. But now the variation over the interior depends on the structure of the system, and in particular on the structure of the interfacial inhomogeneity. It is just because of this dependence of 652 on the structure of the inhomogeneity that we can isolate the thermodynamic properties of the interface through the macroscopic equation (3). In the next section we show how a variation on the boundary of an inhomogeneous fluid sample, for which the reduced grand potential is given by an appropriate van der Waals density functional, may generally be transformed into a variation over the sample interior, and hence how the method of variation under extension may be used to determine the thermodynamic properties of the inhomogeneities as functionals of the density.

3. The variation under extension

of a general van der Waals density functional

Suppose we consider a macroscopic sample of a C-component fluid in which there is a d-dimensional inhomogeneity. That is, the local densities p’, i= 1,2 3 . . 3 C, are functions of d coordinate variables xX, s = 1,2, . . , d. The reduced grand potential of this sample, Q, has the form Q(L) = o,Ld+

o,_,Ld-’

+. . . + 0,L + o,+

G(l/L)

.

(17)

L is a macroscopic length associated with the d-dimensional region S(L) over which Q(L) is defined; (0, is the density of grand potential in the i-dimensional

M. BOITEUX

406

phase

as defined

homogeneity

in a Gibbsian

is ri-dimensional,

AND

thermodynamic

example,

(n-

of the (T-dimensional

d). In the case of the planar

where d = 1 and J=

per unit interfacial

analysis’).

Although

the fluid itself may be &dimensional,

d 3 d; Q(L) is the total grand potential of dimension

J. KERINS

3, 2L is sample

liquid-vapor height; B(L)

fluid per unit “area” interface

taken

P-phase,

bubble

in which the a-phase

as the radius is centered;

(fig. 2), for

is the grand potential

area; w, = - 2p and (II,, = 0’8. For the bubble

d = a = 3. L is conveniently

the inas long as

(fig. 3a) where

of the spherical

volume

of

Q(L) is n”(L), the total grand

potential; oj = 471/3 p”, where p” is the pressure of the bulk /I phase; o> = tc), = 0 and cc)”is the excess grand potential associated with this spherical inhomogeneity. In the case of a straight L is taken

three-phase

as the radius

(fig. 3b), where d = 2 and a = 3,

line of contact

of the cylinder

enclosing

the contact-line

sample;

Q(L)

is the grand potential per unit length of the contact line; (o? = - 7cp. where p is the common value of the pressure in each of the bulk phases (r, fl and y); tc), = @ + afi + 0%;’ is th e sum of the surface tensions and (ti,) = 7 is the line Our aim is to isolate the contribution to Q(L) from the individual o,L’. the variation of O(L) under extension. i =o, 1,. , rf by examining In a van der Waals theory, the reduced grand potential Q(L) is an integral over tension.

S(L)

of a function

dimensional) Q(L)

gradients = L?[p’; L] =

The p’(x) are Euler-Lagrange functional

Y depending VP’. i = 1,2,

on

the densities

. , C; e.g. in Cartesian

1

p’(x)

and

their

(18)

3

and necessarily are solutions of the densities, SQ/6p’(x) = 0, associated with the van der Waals

the extension

652= Q(L + CL) -L?(L)

L + L + CL, from (17) one finds that

= t 1 iw,L'+

C(l/L))

1-I

a

(cl-

coordinates

dx ‘f’(p’(x), BP’(X))

equilibrium equations,

(18). Under

only

(19)

b

Fig. 3. (a) A bubble of r phase inside a b-phase sphere of radius L. (b) A contact line sample enclosed in a circular cylinder of radius L,in which the axis of the cylinder is parallel to the straight three-phase line of contact.

THERMODYNAMIC

so that the contribution

A d-l

=

PROPERTIES

OF INHOMOGENEOUS

of odLd to Q(L)

Q(L)-ljlg

=y

!=I (

may be removed

variation

of A,-,

I

for the present

under extensions

to get A,-,,

0;L’.

l-6

(Here and below we neglect terms of order are of no consequence

407

FLUIDS

l/L in writing

discussion.)

equations;

We could

and remove the contribution

such terms

now consider of od_, Ld-

the

’ from

A N_, to get A,_,, and so on to A, = co,; given the set of d + 1 quantities Sz, , A, we could clearly extract the quantities w,L’, i = 0, 1, . . , d. In A (,- ,, applying potential,

this procedure to the van der Waals functional (18) for the grand we also want to explicitly make use of the fact that the functional is

minimal with respect to variations in the p’(x). Let us write the functional Q[ p’; L], given previously (18) in Cartesian coordinates, in a general curvilinear coordinate system x, where x = (x’, x2, , x”> is the contravariant coordinate vectors),

Q[ pl; L] =

dx ,/m s S(L)

y ( P’>~,P’>cc,,) 3 (21)

4P’h)

W(x) = F.

The gradient of pi, Vp’ = (a,~‘, &p’, . . . , adpi), is a covariant vector. g,,(x) is the metric or fundamental tensor for the given coordinate system (see refs. 8-10 for its properties; any inner product of gradients, such as Vp’ - VP’, depends on the metric tensor). m is the Jacobian of the Cartesian to curvilinear coordinate transformation, and is simply related to the determinant of g,7,, as g(x) = ldet g,s,(x)l. In the general curvilinear coordinate system, we should properly regard the reduced grand potential as a functional of the densities and the metric tensor, i.e. s2 = !2[ p’, g,,; L]. Let us now extend the system by increasing L to L + CL, and then make a change of coordinates, x = x(y, 6) such that S(L + CL), the domain of integration after extension in the x variables, is mapped onto a domain of integration in the y variables that is S(L). For example, in Cartesian coordinates x is related to y by a simple dilation or resealing transformation; indeed the transformation x = x(y, t) represents the simple dilation in terms of the general curvilinear coordinate system. After such a coordinate transformation or resealing, we can evaluate the variation 6Q by a functional Taylor expansion, since the domain of integration for both Q(L) and sZ(L + CL) is S(L). Obviously we must account for the

M. BOITEUX

408

variations

induced

by resealing

632

f2

AND J. KERINS

in both the densities

632 &!?\,(X)&J‘(Y) + . W(x)b’(JJ)

&L(X)W(Y)

and the metric.

-i &

Sp’(x)dp’(y)

.

I

(22) where for the given change

of variables

d.u’(y, c) ?.u’(y. t )

dg:,,(y) =7

~

R,,

8,,’

(x (Y, ( 1)- &,.(Y 1

and

(23) = P’(X(Y? ( 1) - P’(Y)

MY)

We have adopted the summation convention of tensor analysis’), in which any repeated index is to be summed over; this applies to the index of the (contravariant) coordinate vector .Y(s’; s = I, 2, , d) and to the indices of the metric tensor (K,,; s. t = I,?, , d). It also applies to the index i on the densities o’(x), although in this case i is simply an index for the set of the C scalar densities and does not imply any vector properties. Only because the densities are scalar quantities does it make sense to refer to the gradient V/I’. The coordinate transformation

can be expanded

x(y. <) =y

in terms of (

+ Uj(y) + (“(C2).

since for ( = 0, x = y, the identity can then be evaluated k:,,(Y)

= 6 k,,~,v”

6p’(y)

= ~[y1’cQl’]$_

Since these variations

+ R,$,V” C’(f’)

(24) transformation.

+ ‘1”~,gJ = c[q *Vp’]

The variations

+ c’(e)

fig,, and bp’ (23)

.

(25)

+ C’(c?).

6x,,, and 6p’ are of order

(26)

c. we may rewrite

6~ (22) as

(27) But recalling

that Q is minimal

with respect to the variations

in the densities

(i.e.

THERMODYNAMIC 652/6p’(x)

= 0), we

PROPERTIES

OF INHOMOGENEOUS

FLUIDS

409

have

(28)

where T”’ = Ts’(pi, aspi, g,,) is defined in the following equations,

(29) The (contravariant) tensor g” is the inverse of g,,; g”‘g,,.= S:, where S:, is the Kronecker delta. T” is commonly called the momentum or energy-momentum tensor in field theories’.“). Knowing the variations Sg,, (25) and 652 (28) we could calculate A,_, through (20), then consider the variation of Ad_, under extension to get A,_,, and so on. Of course in evaluating 6A,_, it will not be true that 6Ad_ J&pi = 0 (cf. eq. (27) for AC!), but this does not cause difficulty; the method of variation under extension is not limited to functionals which are extremal. Usually in field theories, as Landau and Lifshitz remark”), “. . the metric tensor has no independent significance and the transition to curvilinear coordinates occurs formally as an intermediate step in the calculation of T”“. In the method of variation under extension, however, g,, has assumed a real role in the calculation of 652. We have implicitly assumed that the fluid with a d-dimensional inhomogeneity fills the entire J-dimensional space, although we isolate a finite sample in an imaginary container when defining Q(L) and Q(L + CL). Thus Q[p’, g,Y,;L] and Q[p’,g,,; L + EL] are functionals of the same profiles p’ and metric g,,,, but the former functional is defined on the domain S(L), while the latter is defined on S(L + CL). This is just as in the discussion of the planar interface in section I; the Euler-Lagrange equation (5) for the profile p(x) was independent of L, as were the boundary conditions. We used a coordinate transformation in examining the variation of the grand potential under extension so that s1(L. + CL), like Q(L) itself, was defined on the domain S(L). Q(L + CL) was transformed into a functional over S(L) of the new functions p’ + 6~’ and g,7,+ 6g,,. The variations 6~’ and & in essence map the boundary extension of S(L) to S(L + CL) into a variation over the interior of S(L), and as such have real significance. Or, in less precise but perhaps more familiar terms, when we “compress” our sample from S( L + EL) to S(L), we induce not only a direct local change 6p’(x) in the (density) fields, but we also induce a local “strain” 6g,, in the volume, which is coupled to a local “stress” T”‘. Since the grand potential is extremal with respect to variations in the densities, sZ(L + CL) is insensitive to direct local changes 6~’ and only the

410

M. BOITEUX

“strain-stress”

variations

AND J. KERINS

are important

in 6.0. We discuss

of view further

in section

5.

At this point

the main

steps in the method

clear, and rather we now consider suppose gradient

than pursuing three

a general

specific

that the function form

of variation

prescription

examples.

this mechanical under

extension

functional

are

. d - 1,

for LI,, i = 0, 1,

For each of these examples

Y in the density

point

we will

(20) has a van der Waals

v’(P’. (7,P’.g,,,) = Ii/(P) + ‘.,~(/‘)‘?,p’g,,i’,p~

= Icl(P)+ “,r(Pm’w~

(30)

$(p) and c,JP) are functions of the C number densities in this C-component system. Although the summation convention does apply to the indicesj and k of c(.j,k=

1,2, . ..) C), these indices

do not imply any tensor

character

for the c,~.

i) The struight line of‘ contuct of fhrw .&id phmes’,“) Given an inhomogeneous fluid in which three bulk phases (r, b, y) and three planar two-phase interfaces all meet along a straight line of contact (fig. Id), let us isolate a macroscopic sample of the contact line in a circular cylinder (fig. 3b) of radius L with the axis of the cylinder parallel to the line of contact. Suppose that the Gibbs dividing surfaces, shown as the planes a/?, ~7, and 811 in fig. Id. have a common line of intersection, the contact line, along the axis of the cylinder. The grand potential per unit length of the contact line has the form (with d = 3 and d = 2). (31) where p is the common value of the pressure in each of the three bulk phases. 0 h’ is the interfacial tension of the ~2 interface (K,? = zj?,pp, cc;))and 7 is the line tension.

On the basis of eqs. (20) and (3l),

(32)

A,, = z In the van der Waals

c?[p’; L] =

density

dx J&k s .VL)

functional

for Q(L)

given earlier

in eq. (21)

P’. (‘,P’, g,,) .

with the function ‘P as in (30), the equilibrium profiles p’ depend on the two-component vector x, for which the domain of integration S(L) is a circular

THERMODYNAMIC

PROPERTIES

OF INHOMOGENEOUS

FLUIDS

411

(I

cii3 X2

‘\

/’

/’

X'

mlny

P

I

V

L

Fig. 4. The domain of integration S(L) for x’ and x2 is a circular cross-section of radius L of the contact-line sample. The point (m, n) denotes an alternative location for the contact line associated with an alternative placement of the Gibbs dividing surfaces.

cross section of the cylindrical sample, perpendicular to the contact line and with area XL* (fig. 4). For the present calculation of A, and do, we choose a Cartesian coordinate system (xl, x2) in which the coordinate origin is at the center of S(L). Upon variation of the length L, the appropriate resealing coordinate transformation (24) is simply x(y, 6) = (1 + 6)~ (i.e. q = y) which implies that sg,, = 6gg22 = 2lz) 6gg12 = agg2r= 0. Furthermore,

for !P (30) in Cartesian coordinates,

T1’=$(P)+

(33) it follows from (29) that

cj~[a2pja2pk--alpjalp~,

T” = T2’ = - 2Cjka,Pja2Pk,

(34)

T22= $(P) + cjk[d#jalpk - a2pja2PT. From eq. (28) for a&?,we can then conclude that

1imE= c+o 2c

dx +(p). s S(L)

(35)

This in turn implies that A, =

dx Jgx)Dd~‘,

aspivgrJ

s

(36)

s(L) with Dr = cjk&p’g”‘a,pk = c&”

Vpk .

(37)

With the line tension and one-half the surface tension isolated in A, (32), we now consider the variation of A, under extension, (38)

412

M. BOITEUX

The functional

derivative

AND J. KERINS

of d, with respect

to g,, can be evaluated

Q with d, and Y with D, in (29); for the particular problem,

one can show that (6d,/6g,,)&,

6p’(y) (26) is straightforward

equals

0. Calculation

(recall q = y), as is the calculation

follows from the macroscopic

equation

by replacing

form of D, (37) of the present of the variation of cSd,/Gp,. It then

(32) for d, and (38) for the variation

cid,

that

zz

s

dx (!,[s ‘c,Jp

’ - vp “1 + ?z[_~2c,,~p1. vp “1

(39)

S(L)

The integrand

on the r.h.s. of (39) is clearly a divergence,

and we could transform

C a”‘(L/2) to a contour integral around the boundary of S(L). In such a contour integral with L very large, we can indeed recognize factors associated with the individual tensions ah-’ (cf. the earlier surface tension formula of eq. 12). Finally we can produce, tension r,

on the basis of (32) an integral

expression

s

=

dx s ‘?,[c,kVp ’ * V/j “1

for the line

(40)

s, I , Alternatively.

we could

isolate

the surface

tension

contribution

to iiQ/c in (35)

(41)

and

then

deduce

from

(32) and

(36) the more

familiar

formula

(in Cartesian

coordinates)

=s dx

~,J’P’.~P’-

iti

+p])

(42)

.S(71

The two expressions for T, (40) and (42) are equivalent as can be shown by using the fact that the equilibrium densities p’(x) satisfy the Euler-Lagrange equations associated with the density functional Q[p’; L] (i.e. the equations ?Q/6p’ = 0). Eq.

THERMODYNAMIC

PROPERTIES

OF INHOMOGENEOUS

FLUIDS

413

(42) is particularly useful in the numerical computation of the line tension r *,“). Our assumption that the Gibbs contact line coincides with the axis of the cylinder enclosing the three-phase sample is arbitrary. Suppose that we had a different placement of the Gibbs dividing planes and line, one where the contact line intersects the cross section S(L) at the point (m, n) (fig. 4). With this choice we must write a more general surface tension sum c oK’hKiin (31); h””is the length of the ~1 interface in S(L). Although the lengths h”’ are functions of L, m and n, under the extension of L to L + CL, the variations 6h”” are (43) Given this behavior (43) for the variations, there is essentially no difference in the calculation of d, and d,. But then in eq. (39) or (41) we have an (asymptotic) integral expression for the surface tension sum, which is clearly independent of the choice (m, n). This implies that 0 = lim 1 oK’:L- 1 o”h”(L, m, L-a

n)],

(44)

[

independent of m and n. Eq. (44) is equivalent to the well-known Neumann triangle conditions for thermodynamic mechanical equilibrium at the contact line’2?. ii) The contact line between two surface phases4,13) Let us consider the meeting at a straight contact line of two distinct structures for the planar interface between bulk phases c( and b (fig. 5). For example, we could imagine an interface at which one of the species is strongly adsorbed in contact with an interface at which the same species is weakly or even negatively adsorbed. A macroscopic sample is isolated in an imaginary rectangular cylinder of width L, height L’ and length L”, whose axis is parallel to the line of contact. In a Gibbsian thermodynamic analysis of this system, there is one dividing plane (shown at height x; in fig. 5~); this plane is separated into two sections by a straight line of contact. The section to the left of the line has area L”w, 0 < w < L and is associated with the first (I) interface structure of tension a;fi; the right-hand section has area L”(L - w) and is associated with the second (II) interface structure of tension rr$“. If the common value of the bulk-phase pressures is p, then the excess grand potential per unit length is a(L) Q(L) =

QL, L’, L”) + pLL’L” = 0;‘L + [(0$ L”

O;')W + 51,

(45)

where T is the line tension of the interfacial contact line. Although the inhomogeneity in this case is two-dimensional, d = 2, by removing the bulk-phase pressure beforehand in 0 we have effectively fixed o2 = 0. The density functional

M. BOITEUX

414

AND

J. KERINS

b

Fig. 5. (a) The straight line of contact between structures I and II for the planar I[{ interface; (b) A sample of the two-phase system enclosed in a rectangular cylinder, whose axis is parallel to the straight line of contact: (c) the interfacial Gibbs dividing plane at height _r’,.

for a is still that of (21) except that in the function Y (30) which appears in the integrand of Sz[ p ‘, L] we should replace tj (p) by J(p) = $ ( p) + p. If we choose a Cartesian

coordinate

the x’-axis

parallel

perpendicular

system

to the contact

.?-direction (with the _r2 = x’/(l + L)) we have 6g22 = 2t , Applying

interface

is perpendicular

in a cross-section

line (fig. 5c), then in the variation

subsequent

resealing

behavior

A, = a;“L =

equations

and

S(L, L’) which

transformation

of Q(L)

is

in the

J’ = .Y’ and

Sg,, = 6g,: = sgz, = 0.

the appropriate

thermodynamic

(.u’, _v~) in which the xi-axis

to the planar

(46) of variation

(45) we conclude

d.v r&p) + ‘;J&P’?,P~

(28) and

(29) for the given

that

- &‘??pk]

(47)

and

A0 = [(a;/ - (T;‘$v + T] =

s

dx 2c,,&p j&p k .

(48)

S(L.L’)

Now we observe O(L)

= o$L

that we could have written +

[(@ - &)W + t]

)

the thermodynamic

formula

(45) as (45’)

THERMODYNAMIC

PROPERTIES

OF

INHOMOGENEOUS

FLUIDS

415

where M”= L - w has the same degree of arbitrariness as w. We would then find the same expression (47) for d,, but now equal to [T$L. This leads to the conclusion that opfl= o$, a necessary condition of thermodynamic equilibrium for the interface, and hence to the conclusion that d, = z. iii) Spherical bubble As the last example in this section, we wish to consider a bubble of cc-phase located at the center of a spherical sample of b-phase, in which the sample radius is L (fig. 6). In this case L?= d = 3 and !2(L) is the total grand potential a(L),

where p” and pB are the pressures in phases a and p, R is the radius of the Gibbs dividing surface, and r~x0 is the interfacial tension. In order to make connection with some earlier work’4.‘5) on th e van der Waals theory of spherical interface, we choose a spherical coordinate system (fig. 6) x’ =r,

x2= 4,

x3=0

(50)

and restrict ourselves to a one-component functional (21) may be written

system. Then the van der Waals

(51) In examining transformation

the variation of G? under extension in L (with the resealing affecting only the radial variable r) we have

&L, = 2&V, &,, = 0,

Fig. 6. The spherical bubble is surrounded

s z t, s, t = 1, 2, 3 ,

coordinate system by fl phase.

for the spherical

(52)

sample

of radius

L in which

an a-phase

416

M. BOITEUX

which leads from (28)

AND J. KERINS

(29) and (49) to the conclusion

that

(53)

We can further

deduce

that

(p!‘-p’)$R’+rr”4nR’ 1 Thus

we have a simple

integral

expression

for the contribution

to 52 from

the

bubble-inhomogeneity, both the interior a phase and the spherical c$ interface. At the given T and p for this one-component system, the phase /I exists as a bulk (stable or metastable) equilibrium phase and so has a well-defined pressure p”( T, p). In a Gibbsian thermodynamic analysis, p’ is the pressure of a homogeneous reference phase c( with the same T and p as the bulk p phase, but in general there is no clear thermodynamic prescription for explicitly evaluating p’; in the present van der Waals theory, however. both p’ and p” are determined by the function Ic/(p). Accordingly, the choice of the reference pressures p” and p ii is independent of the radius R of the Gibbs dividing sphere. If we differentiate (54) with respect to R we find the thermodynamic Laplace equation’.‘“) (55) since the r.h.s. of (54) is independen write

‘55) in (54) we may

(56)

Only the particular combination of p’, p”, cXp and R given on the 1.h.s. of (54) equals the total excess grand potential associated with the bubble. and as such is independent of R. The arbitrariness as to how this total excess is assigned to the interior reference phase a or the spherical c$ interface leads to a formal dependence of the surface tension (56) on the choice of R. If we choose R,, as the radius where [daXB/dR] = 0, then (55) reduces to the classical Laplace equation of capillarity and R,, is called the radius of the surface of tension. In this case a’0

THERMODYNAMIC

and R, are unambiguously

PROPERTIES

determined

OF INHOMOGENEOUS

417

FLUIDS

by (55) and (56)

and 4 I$ dr r2c( P)(dp /dr)2 %=

(P-P9

1 “3

(57b)

Thus we have simple integral expressions for the radius of the surface of tension and the surface tension at that radius, which depend only on the reference states LXand /3 and on the equilibrium density profile p(r). These three examples demonstrate how the technique of variation under extension can be applied in density functional theories of the van der Waals type to deduce nontrivial functional expressions for the thermdoynamic properties of the inhomogeneities, e.g. (42), (48) or (54). Moreover, certain conditions of thermodynamic equilibrium, e.g. (44) or (55), follow directly from the invariance of functional expressions to the choice of Gibbs dividing surfaces. In the next section we extend the method of variation under extension to a particular class of nonlocal density functionals.

4. Nonlocal

density functional

We now examine the variation under extension of the reduced grand potential 52 of an inhomogeneous fluid sample in the case that C! is given as a nonlocal functional of the equilibrium densities p’(x), Q[P’; Ll

= dx dv &it%?i%,d~~~)~‘(-W(y) s

s

+

s S(L)

dx Jg(x)W.

(58) Here h,,(x, y) are functions

of x and y, and Ic/(p) is a function

of the densities

p’.

Such nonlocal density functionals appear, for example, in the mean-field theory of penetrable-sphere models3.12), and in the theory of fluids with long-ranged pair potentialsI 19). ‘/I(P) is that part of the local density of grand potential which depends only on the densities pi at X. Depending on the model it might be the grand potential of an ideal gas3.12), or of a reference hard-sphere systemI 19). The interaction of fluid particles at x with those at y is responsible for the nonlocal interaction kernels hj(x, y). The domain of integration for x and y in the first term on the r.h.s. of (58), the nonlocal piece, must be specified. The point x may be anywhere in the region S(L)

418

M. BOITEUX

which is determined

by the shape of the macroscopic

simply one portion. which occupies

AND J. KERINS

isolated

the entire

in an imaginary cj-dimensional

with fluid particles

assume,

that the values of the interaction

if x and y are greater neighborhood

than

of radius

at points

a distance

< of x ES(L)

Since the sample is

of an inhomogeneous

space, fluid particles

may still interact however,

sample.

container,

at points

s. which lie outside

5 apart.

kernals Thus

fluid x ES(L)

of S(L).

We

/z,~(x,J) are negligible only

need be considered,

points

y within

and the domain

a of

integration for y in (58) may be restricted to the region S(c) which contains S(L) but is larger in extent by a factor of 4, i.e. t = L + [. When we examine the variation of R(L) under extension, we implicitly will be varying both S(L) and S(E). Note also that the equilibrium profiles p’(x) necessarily satisfy the Euler-Lagrange equations associated with the functional (58) in the limit of infinite sample size. That is, for any x in S(-,), where S(x) is the limiting region for S(L) in the limit that L goes to CC. 0~

a*

lx2

]im -==+ &l’(x)

IT.L-rL

&7’(x)

s

dy h(x>y)

+ ~,(~.Y)IP~(Y),

subject to the appropriate boundary conditions at infinity. i.e.. on the boundary of S(a). For this reason the variation of the functional Q(L) of (58) with respect to the densities p’(x). XES(L) (60)

is not quite 0 because of the finite or truncated domain S(L). Nevertheless, in all cases below we will set SQ/6p’ to 0 since the truncation error, which originates in the difference between S(L) and S(x), vanishes in the limit that L and E go to CD. In a general actually kernals

curvilinear

a functional

coordinate

of the densities

system, the nonlocal functional (58) is the metric tensor g,, and the interaction p’,

hiA;

Q(L) = Q[P’. g,,, h,r;

Ll

(61)

When examining the variation of C?(L) under extension in L, we must account for the variations 6~’ in the densities, fig,, in the metric and 6h+ in the interaction kernals. Thus (cf. eq. 22), 6Q

. -----p'(x)+ W(x)

"nk,Wj+j &v(x)

.%I., x

SC/~) +

higher-order

dx

terms in the variations.

j

dy

{~6h,(v)j

XL)

(62)

THERMODYNAMIC

PROPERTIES

OF INHOMOGENEOUS

FLUIDS

419

As always, the grand potential is insensitive to local variations in the density since &2/6p’= 0, so the first term on the r.h.s. of (62) vanishes. The earlier formulas still apply for evaluating the functional derivative with respect to the metric (29) and for the variation in the metric (25). The variation 6hjk can be calculated for the earlier resealing coordinate transformation (24)

(63) and the functional derivative &2/6hjk is also available g

=

&b”(Y).

(64)

/k

We may now write the final result, based on (62), in the form limE= c+o t

dx dY ,/&%(%‘(x)~~(v) s s S(L) X S(L) -Mx) + ~MY) axs ays

+

+

[11 (x ) ’ V.&j/c + ‘I (Y) ’ V&d

s dx

1

J~(X)J&)PW(X) +gT%gtsl~

(65)

S(L)

With this result we could calculate A,_,, then determine A,_, and so on. Again though, rather than proceeding with a general formalism, we turn instead to four specific examples. i) The planar liquid-vapor interface in a van der Waals fluid Let us imagine a one-component two-phase system with a planar interface (fig. 2), in which the individual fluid particles interact pairwise with a potential 4(r) of the type proposed by Kac”) (for C?= 3)

M. BOITEUX

420

If we adopt

the properly

scaled coordinates

limit of small y, the grand (approximate)

functional

potential

x = YYand take the (van der Waals)

per unit area Q(L)

for this system

has the

form

L

YQ[P, Ll=

AND J. KERINS

/.

s

L

d.r e -1’ %(.‘oP(.r) -L

I

(67)

IT

with the Cartesian coordinate .Yas in fig. (2). Note that the length L has also been scaled by the inverse length y, and we have implicitly assumed E > L. The dependence that

of Q(L)

on L is known

(I), and so we deduce

from (65) with rl=

1

(68a)

where h(.r, J*) = - 4~ e Ii ‘1. We may rewrite

From

this integral

formula

this as

for the surface tension, we can reasonably argue that only for 1-v~ ~‘1< 1, but for e-i’- ‘1 is appreciable

0”’ > 0. The exponential 1.~-.I/ < I the entire integrand

in (68b) is positive and hence we expect gXxiito be to positive. Indeed in the strict limit ;‘+O. CT”’diverges to infinity corresponding the infinitely long-ranged attractive potentials”), but this divergence does not concern us here. ii) T/w planur inte[f:fuce in u tlt‘o -c’on~ponent penerruhk -.vplwc tmm’~~i We consider a two-component system with species 1 and 2, in which particles of the same species do not interact while particles of different species interact as hard spheres of radius 1. This is the primitive version of the penetrable sphere model introduced by Widom and Rowlinson”). For a given range of chemical potential ,L /L = ,u’ = ,u’, coexistence of an r-phase, rich in species 1. with a P-phase. rich in species 2, is possible. The structure and thermodynamics of a planar interface between these phases has been extensively studied by Rowlinson et al.“). For a two-phase system with a planar interface (fig. 2) the grand potential

THERMODYNAMIC

PROPERTIES

OF INHOMOGENEOUS

per unit area Q(L) in a mean-field approximation

--=

1kT

has the functional

421

form2)

L+I

L

w,)vo

FLUIDS

dx

s

-L

+

s

-CL+

dy hi& Y)P’(x)P’(Y) I)

1dx ,W)b(d(x)lJ)

- p’(x)1,

(69)

with h,, = hz2= 0 and

h&3y)= hA%Y)= i

lx-Yl>L

0, $1

_

(x

_y)‘],

(x

_

y(

<

1

.

(70)

J. is the common value of the activity for species 1 and 2. The activity 2 and the densities pi and p2 are all dimensionless, having been scaled by the volume r0 = 47r13/3;the length L and coordinate x are also dimensionless, having been scaled by the hard-core diameter 1. Applying the method of variation under extension to the functional (69) just as it was applied (67) in the van der Waals fluid, we find

x+=s s d%l

dx

dy h,(x> y)[p’(x)p’(y)

+

P~(x)PY.Y)I 7

(71)

where

0, h,(x,y)=

i $3(x -1’)l-

lx-,++ 11, IX -yJ < 1 .

(72)

The formula (71) for a@ is identical (under the approximate transcription of the two-component primitive model to the one-component penetrable-sphere model) to the surface-tension formula derived, in a mean-field approximation, by Leng et a1.z3)directly through the canonical partition function. This is not surprising since the method of variation under extension is not unlike the method employed by Leng et a1.23) of scaling the linear dimensions of the system in order to differentiate the canonical partition function with respect to the interfacial area. On the other hand, the agreement between the surface-tension formula of Leng et a1.23)and (71) may be due to the high degree of internal consistency of the present mean-field approximation for the mode122). It is not obvious that the surface tension derived by a differentiation of the canonical free energy with respect to the interfacial area in a mean-field approximation will be identical to the tension derived by the method of variation under extension applied to a mean-field density functional.

M. BOITEUX

422

iii) The three-phase

line qf‘rontact

In a three-component coexisting

phases

respectively, circular

(c(, /I, y), each

section

in a three-component penetrable-sphere

penetrable-sphere

meet along

cross

AND J. KERINS

of which

a straight

with radius

model,

it is possible

is rich

in the species

line of contactj)

line (figs. 3b, 4) then the excess grand

contact

line, a(L)

= Q(L) +pxL’.

sample

potential

per unit length

has the thermodynamic

line

of the

form (cf. eq. 31) (73)

where chi is the interfacial r is the

is a

of the three-phase

Q(L)+J”‘Lt_z.

and

three

i, i = 1,2. 3.

as in fig. (2d). If S(L)

f. of a cylindrical

contact

model

to have

tension

tension of the

of the planar contact

tij. interface

line.

The

(tij, = L$, 87 or r*)~)

straight

contact-line

in-

homogeneity is two-dimensional (ri = 2). but by considering the excess potential o(L) where the pressure contribution is already removed we have fixed W? = 0. The mean-field density functional for this contact line system is

dy h,, (13 y )u’(x )u

‘(y ) +

XI.1 Y S(I.1

d-d(p). s S(l.1

(74)

The equilibrium densities p’(x) (i = I, 2, 3) depend on the position x in S(L), and along with the common value of the activity 2, have been scaled by the volume z’(,= 47#/3; the length L and coordinates x have been scaled by 1. The interaction kernels h,,(x. y) are zero if i =,j. and if i #,j they depend only on z = IX -yl in the form 0, q&Y)

= k,(=)

The function

g(p)

g(p)

=

3 &I

:>I, (75)

of the densities

= P’(x)[ln(p’(x)/i)

-I < 1

-?)“?,

is

- p’(.u)] + $T.

(76)

where p is the common value of the pressure in each of the bulk applying the method of variation under extension we find

xa”“L

=

dx s s S(L) X SC/.+ I)

i i ,:lk=l i#h

P’(x)P”(Y)

1s +

phases.

dx 2$(p),

SC L)

On

(77a)

THERMODYNAMIC

where, with z = IX -

PROPERTIES

OF INHOMOGENEOUS

FLUIDS

423

y(,

(77b)

and

h,(x, y) = h”(Z) =

3 &2;:;;,,2]l

z < ‘.

Note that h,(i) and h,(i), although integrable, are divergent at z = 1; this is a reflection of the underlying hard-sphere interaction. Thus in the functional (74) for the excess grand potential we have distinguished the surface tension contribution

(77) from the line tension

contribution

(78).

iv) Fluid ugainst a solid wall We now generalize our earlier definition of an inhomogeneous the case of a single-component fluid next to a solid. We consider

fluid to include the case in which

the solid-fluid interface is planar, with the fluid density p depending only on the coordinate .Y measured perpendicular to the interface (fig. 7). Furthermore, we model the solid as a rigid, impermeable continuum of density p.!, which fills the entire half-space x < 0. We isolate a macroscopic sample of this solid-fluid system in an imaginary rectangular prism such that the solid-fluid interface is parallel to

i

Fig. 7. The planar solid-fluid interface

M. BOITEUX

424

a pair of opposite 21,. The grand naturally

AND J. KERINS

prism faces of area A and also bisects four prism edges of length potential

per unit

interfacial

area

Q,(L)

of the sample

splits

into two pieces (79)

Q,(L) = Q,,,,,,(L) + Q(L) Q,,,,,,(L) is the contribution

to Q,(L) from the solid in the half of the sample lying

in the region .v < 0, and Q(L) is the contribution the sample in the region .Y > 0. For our purposes, supposed to include any contributions induced by the model solid; Q(L)=

-pL

to Q,(f,)

from the fluid in that half of only Q(L) is of interest and is from

fluid

inhomogeneities

+o.

(80)

where p is the pressure in the bulk fluid and 0 is the solid-fluid interfacial tension. We assume that the interactions of the solid with the fluid are represented by an external field U,(s) acting on the fluid in the region I > 0. The equilibrium density functional for the sample can be written as i

/ C’(L) =

ds I 0

if we assume

/ d.r /I(\-, .r)p(.~)p(~)

s 0

+

d.x G(p) + j_ 0

that the fluid-fluid

interactions

ds U,(s)~>(s)

.

(81)

s /I are well-represented

by the first two

terms on the r.h.s. of (81). For example, in a van der Waals fluid Ii/(p) is the density of grand potential in a hard-sphere reference system and 11(x. .I’) is related to the long-ranged attractive potential’“). Q is now a functional of the density /I(.\-). the interaction kernel Iz(.Y,,I.) and the external field due to the solid U&s). Accordingly, in the variation of Q(L) under extension in L, there will be a term arising from the variation of the external field after the appropriate coordinate transformation.

+

s ds--

6Q

SU,(s)

(SU,(s) + higher-order

where, as always, 652/6p(s) known (63. 64), and

terms in the variations,

= 0. The term in (82) from the variation

(82)

in h(.~, JS) is

(83)

THERMODYNAMIC

PROPERTIES

On the basis of the thermodynamic method

of variation

under

OF INHOMOGENEOUS

equation

extension,

(SO), we can isolate,

the solid-fluid

interfacial

0

0

.i0

through

tension

h(x,y)+;+yE + dxp(x) 8.~1

425

FLUIDS

-xdU’ [

the

0,

.

dx]

(84) From (84) there are clearly distinct contributions to (T from the fluid-fluid interactions and the solid-fluid interactions. Suppose we choose a van der Waals fluid (6668) near a van der Waals solid19) for which U,(x) = -c,

ee‘ .

(85)

In that case we have (cf. eq. 68) 3c

d_v (1 -(x

dx

ya =;

s II

X

3( -Yl)e-I”-~lp(y)p(x)-~,

s

dxx

e-‘p(x).

(86)

s

0

0

Thus, the solid-fluid interactions make a negative contribution to the tension (T, as expected because of the attractive nature of U,. And depending on the magnitude of c, relative to Q, we might easily calculate negative solid-fluid tensions24). We must remark that a negative solid-fluid tension is possible only because of the rigidity of the solid. Consider an interface of tension 0 and with area A in a two-phase equilibrium system in which each bulk phase has a fixed volume. A variation to a nearby equilibrium state, which increases the interfacial area to A + 6.4, must be accompanied by a rearrangement of the material in both of the bulk phases adjacent to the interface, in order to satisfy the constraints of fixed bulk-phase

volume.

Accordingly,

for the supposed

variation,

the total

grand

potential of the system will increase by an amount aL5A due to the interfacial variation and by an amount 6 W due to the necessary rearrangement of the bulk phase materials. Now the material in a fluid phase is so mobile and malleable that no work is done in rearranging the fluid and 6 W = 0 for a bulk fluid phase. In a system of two fluid phases, a variation 6A in the interfacial area causes a variation oaA in grand potential of the system; thus a necessary condition for a stable fluid interface is cr > 0. On the other hand, the material in a bulk solid phase is so rigid that a rearrangement of the solid requires stress-strain work with 6 W > 0. At a solid-fluid interface a variation of 6A in the interfacial area causes a variation o6A + 6 W in the grand potential. The necessary condition of stability for the solid-fluid interface is that adA + 6 W be positive for variations from our

M. BOITEUX

426

initial equilibrium solidNluid

state. Clearly

tensions

AND J. KERINS

this does not imply 0 > 0 in general,

are thermodynamically

and negative

allowed.

5. Discussion In the density variation dynamic

functional

formalism

of inhomogeneous

fluids,

the method

of

under extension is useful for expressing the macroscopic, thermoproperties of the inhomogeneities as functionals of the microscopic, local

densities. In particular, the microscopic expression for the excess grand potential associated with an inhomogeneity follows from a comparison of the variation of the (reduced) grand potential Q(L) under extension in L as computed from a thermodynamic equation (17) with the variation computed from a densityfunctional equation (65). The key step in computing the variation of the density-functional expression for Q(L) is a transformation of coordinates after which the extended system is defined over the same domain of integration as the original unextended system. Although we have restricted our attention to the excess grand potential, the other thermodynamic properties of the inhomogeneity can be generated by differentiation of the excess grand potential with respect to the thermodynamic parameters T and p’, i = 1,2, . . , C (cf. eq. 16, the Gibbs adsorption equation for a planar interface). The method of variation under extension is very general and may be applied to functionals other than Sz. For example, the method could be applied to the functional N’[p’, L] for the (reduced) number of particles of species i in S(L) N’[ p’; L] =

dx p ‘(x )

(87)

to get adsorptions directly. method as presently applied space (i.e. the thermodynamic

The single physical assumption necessary for the is that, although the inhomogeneous fluid fills all limit has already been taken), we can focus our

SC/. ,

attention on a large but finite-sized sample, in which the inhomogeneities are located. Therefore in the thermodynamic equation (17) for L?(L) we allow for terms of order (l/L) due to the finite size of the sample, but we need not allow for other terms which would be associated with edge effects if the fluid itself filled only a finite region of space. The method of variation under extension may also be applied to a much broader class of density functionals than those considered above in sections 3 and 4. For example, the method could be applied to density functionals of the type devised by Nordholm et al.25,‘6) on a phenomenological basis, or to functionals of the type proposed by Percus”) on the basis of exact results for one-dimensional fluids. In

THERMODYNAMIC

both cases the grand p(x),

potential

but also on a function

The method Ebner,

of the density

As a final point,

OF INHOMOGENEOUS

is functionally

in that case the interaction

421

FLUIDS

not only on the density of the density

to the approximate-density-functional

p(x).

model

of

kernel h(x, y) is itself a

p(n).

we note that our approach

extension of the density analysis of deformation

dependent

p(x) which itself is a functional

is also applicable

Saam and Stroud2’);

functional

PROPERTIES

for evaluating

the variation

under

functional Q[p’; L] has a classical antecedent in the in continuous media29). Let us associate the original

sample of inhomogeneous fluid with a continuum of material in an initial or undeformed state which occupies a region S(L) of space. After the appropriate resealing

coordinate

transformation

(24) we may associate

the extended

sample

with the same continuum of material occupying the region S(L) but now in a final or deformed state’“). A pointy in the original sample is shifted to the new point x = x(y, t) associated

under with

deformed states. density functional transport

the

deformation.

a distortion

A variation

of the volume

in the

element

metric

between

tensor

g,, is

the initial

and

Moreover, our calculation of 652[p’; L], the variation of the under extension, may be regarded as an application of Reynolds

theorem”),

6Q[p’, L] = 6 j+dU’(y)= Xl.)

1 dy W’ + ‘f’(V.~(y))l,

(88)

.-WI

with

for a variation feature

with respect to 6 of the local density of grand potential

central

macroscopic, macroscopic microscopic

to the method thermodynamic and microscopic

expressions

of variation equation equation

under

extension

(17) for Q. Only because for O(L)

for the thermodynamic

Y. The novel

is the additional we have both a

are we able to derive

properties

simple

of the inhomogeneities.

The application of the method of variation under extension to an exact density functional which involves the direct correlation function’.‘4.‘2), and the application of the method directly to the grand partition function are currently under investigation”). In these exact microscopic theories for the grand potential Q, the stress-strain interpretation of the variation 652 can be more precisely discussed in terms of the microscopic stress tenso@). The mechanical interpretation of 652 as stress-strain work has its limitations, as recently pointed out in the case of a spherical interface34,35). But whether the microscopic expression for Sz comes from an exact statistical-mechanical theory or an approximate density functional theory, the variation 6Q can be calculated unambiguously on the basis of the

428

M. BOITEUX

macroscopic,

thermodynamic

dent of any detailed

equation

microscopic

is therefore

independent

microscopic

interpretation.

of any

AND

J. KERINS

for Q. Hence the variation

interpretation,

652 is indepen-

such as stress-strain

of the difficulties

associated

with

work,

and

a detailed

Acknowledgements We wish to acknowledge support from the Centre National de la Recherche Scientifique (M.B.) and the U.S. Department of Energy (J.K.). We are grateful to Professors H.T. Davis, M.E. Fisher, L.E. Striven and B. Widom for critical comments. We also appreciate several useful remarks on the spherical interface by A.H. Falls.

References I) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28)

J.W. Gibbs, Collected Works (Longmans and Green, New York, 1928) Vol. I, pp. 55-353. J. Kerins and B. Widom, J. Chem. Phys. 77 (1982) 2061. R. Evans, Adv. Phys. 28 (1979) 143. J. Kerins and M. Boiteux, Physica 117A (1983) 575. B. Widom in: Statistical Mechanics and Statistical Methods in Theory and Application, U. Landmann, ed. (Plenum, New York, 1977) pp. 33371. H.T. Davis and L.E. Striven, Adv. Chem. Phys. 49 (1982) 357. L. Boruvka and A.W. Neumann, J. Chem. Phys. 66 (1977) 5464. IS. Sokolnikoff, Tensor Analysis Theory and Applications (Wiley, New York. 1951). D.E. Soper, Classical Field Theory (Wiley, New York, 1976). L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1975) fourth revised English edition. Ref. IO, p. 273. J. Kerins, Ph.D. Thesis, Cornell University, 1982. J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity (University Press, Oxford. 1982). A.J.M. Yang, P.D. Fleming and J.H. Gibbs, J. Chem. Phys. 64 (1976) 3732. A.H. Falls, L.E. Striven and H.T. Davis, J. Chem. Phys. 75 (1981) 3986. N.G. van Kampen, Phys. Rev. A 135 (1964) 362. C. Varea, A. Valderama and A. Robledo, J. Chem. Phys. 73 (1980) 6265. A. Robiedo and C. Varea, J. Stat. Phys. 26 (1981) 513. D.E. Sullivan, J. Chem. Phys. 74 (1981) 2604. M. Kac, Phys. Fluids 2 (1959) 8; M. Kac, G. Uhlenbeck and P.C. Hemmer, J. Math. Phys. 4 (1963) 216. B. Widom and J.S. Rowlinson, J. Chem. Phys. 52 (1970) 1670. J.S. Rowlinson, Adv. Chem. Phys. 41 (1980) 2 and references therein. CA. Leng, J.S. Rowlinson and S.M. Thompson, Proc. Roy. Sot. A352 (1976) I; A358 (1977) 267. G.F. Teletzke, Ph.D. Thesis, University of Minnesota, 1983. S. Nordholm, M. Johnson and B.C. Freasier, Aust. J. Chem. (to be published). M. Johnson and S. Nordholm, J. Chem. Phys. 75 (1981) 1953. J.K. Percus, J. Chem. Phys. 75 (1981) 1316. C. Ebner, W.F. Saam and D. Stroud, Phys. Rev. A 14 (1976) 2264.

THERMODYNAMIC

29) We are grateful 30) 3 I) 32) 33) 34) 35)

to Prof.

PROPERTIES

L.E. Striven

OF INHOMOGENEOUS

for pointing

out the connection

FLUIDS

between

the method

429

of

variation under extension and Reynolds transport theorem. Ref. 8, Ch. 6. R. Ark, Vectors, Tensors and the Basic Equations of Fluid Mechanics (Prentice-Hall, Englewood Cliffs, 1962) p. 84. W.F. Saam and C. Ebner, Phys. Rev. A 15 (1977) 2566. J. Kerins (unpublished). P. Schofield and J.R. Henderson, Proc. R. Sot. Lond. A 379 (1982) 231. S.J. Hemingway, J.R. Henderson and J.S. Rowlinson in: Structure of the Interfacial Region, Faraday

Symp.

No. I6 (Royal

Society

of Chemistry,

London,

1982)

pp. l-l I.